Information Processing Letters 110 (2010) 913–916 Contents lists available at ScienceDirect Information Processing Letters www.elsevier.com/locate/ipl Decomposition of sparse graphs into two forests, one having bounded maximum degree Mickael Montassier a,∗,1 , André Raspaud a,1 , Xuding Zhu b,c,2 a Université Bordeaux 1, LaBRI UMR CNRS 5800, France b Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung, Taiwan c National Center for Theoretical Sciences, Taiwan article info abstract Article history: Received 14 March 2010 Received in revised form 18 May 2010 Accepted 10 July 2010 Available online 17 July 2010 Communicated by B. Doerr Keywords: Combinatorial problems Decomposition Forest with bounded degree Discharging procedure Global rules Let G be a graph. The maximum average degree of G, written Mad(G), is the largest average degree among the subgraphs of G. It was proved in Montassier et al. (2010) [11] that, for any integer k  0, every simple graph with maximum average degree less than m k = 4(k+1)(k+3) k 2 +6k+6 admits an edge-partition into a forest and a subgraph with maximum degree at most k; furthermore, when k  3 both subgraphs can be required to be forests. In this note, we extend this result proving that, for k = 4, 5, every simple graph with maximum average degree less than m k admits an edge-partition into two forests, one having maximum degree at most k (i.e. every graph with maximum average degree less than 70 23 (resp. 192 61 ) admits an edge-partition into two forests, one having maximum degree at most 4 (resp. 5)).  2010 Elsevier B.V. All rights reserved. 1. Introduction All considered graphs are simple. A decomposition of a graph G is a set of edge-disjoint subgraphs whose union is G . By the well-known Nash-Williams’ Theorem [12], ev- ery planar graph decomposes into three forests. Balogh et al. [1] conjectured that one of the three forests can be required to have maximum degree at most 4, which is sharp infinitely often. They proved several results in this direction, and Gonçalves [7] proved the full conjecture. In addition, he showed that planar graphs with girth at least 6 (at least 7) decompose into two forests with one having maximum degree at most 4 (at most 2). From his proof, one can easily derive that every graph with maximum av- erage degree less than 3 decomposes into two forests, one having maximum degree at most 4. We recall that the * Corresponding author. E-mail addresses: montassi@labri.fr (M. Montassier), raspaud@labri.fr (A. Raspaud), zhu@math.nsysu.edu.tw (X. Zhu). 1 Research supported by French–Taiwanese project CNRS/NSC – New trends in graph colorings (2008–2009) and the ANR project GRATEL. 2 Research supported by NSC97-2115-M-110-008-MY3. maximum average degree of a graph G , written Mad(G), is the largest average degree among the subgraphs of G . Previously, Borodin et al. [6], He et al. [9], and Kleitman [10] were interested by decomposing planar graphs (with large girth) into a forest and a subgraph with bounded maximum degree: every planar graph with girth at least 9 (resp. 6, 5) decomposes into a forest and a subgraph with maximum degree at most 1 [6] (resp. 2 [10], 4 [9]). Such decompositions have applications for the game color- ing number; see [13]. In this perspective, decompositions of a graph with given maximum average degree into a for- est and a subgraph with bounded maximum degree were stated in [11]. More precisely it was proved that, for any integer k  0, every simple graph with maximum average degree less than m k = 4(k+1)(k+3) k 2 +6k+6 decomposes into a forest and a subgraph with maximum degree at most k; further- more, when k  3 both subgraphs can be required to be forests. This note is a follow up of this work. We prove that: Theorem 1. For k = 4, 5, every simple graph with maximum average degree less than m k = 4(k+1)(k+3) k 2 +6k+6 decomposes into two forests, one having maximum degree at most k. 0020-0190/$ – see front matter  2010 Elsevier B.V. All rights reserved. doi:10.1016/j.ipl.2010.07.009